Research output: Contribution to journal › Article › peer-review
Journal | International Mathematics Research Notices |
---|---|
Date | Published - 2010 |
Issue number | 1 |
Volume | 2010 |
Number of pages | 18 |
Pages (from-to) | 69-86 |
Original language | English |
Let A(n,m)(psi) denote the set of psi-approximable points in R-mn. Under the assumption that the approximating function psi is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.
Project: Research project (funded) › Research
Project: Research project (funded) › Research
Project: Research project (funded) › Research
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